Causal and Bayesian Network(Chapter 2)

Book: Bayesian Networks and Decision GraphsAuthor: Finn V. Jensen, Thomas D. Nielsen

CSE 655 Probabilistic ReasoningFaculty of Computer Science,

Institute of Business Administration

Presented byQuratulain

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OutlineReasoning under uncertainty

Causal network and d-separation

Bayesian networkGraphical model

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Reasoning Under UncertaintyWhy Reason Probabilistically?In many problem domains it isn't possible to

create complete, consistent models of the world.

If information is given with certainty then Propositional logic (Truth table) can be used.

Want to make rational decisions even when there is not enough information to prove that an action will work.

To deal with uncertain events, we extend truth value of propositional logic to certainties which are number between 0 and 1.

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Example (Type of reasoning that human do daily)

“In the morning, my car will not start.”Reasons:

◦I can here starter tune, so must be power in battery

◦May be fuel has been stolen overnight◦The spark plug are dirty◦May be due to the dirt in carburetor◦A loose connection in ignition system or

any thing serious

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A Causal Perspective – Car Example Construct a graph to represent

causal relationship between events which gives structure to the situation for reasoning.

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Variable (node)

States

Fuel {yes, no}

CleanSparkPlugs

{yes, no}

Fuel Meter {full, half , empty}

Start {yes, no}

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OutlineReasoning under uncertaintyCausal network and d-separation

Bayesian networkGraphical model

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Causal network and d-separationA causal network consists of a

set of variables and a set of directed links between variables.

Mathematically, the structure is called a directed graph.

Causal networks can be used to follow how a change of certainty in one variable may change the certainty for other variables.

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3-Cases of evidence transmitionSerial Connections

Diverging Connections (common cause)

Converging Connection (common effect)

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P(C|A^B)=P(C|B)

P(C|A^B)=P(C|B)

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Serial ConnectionsEvidence about A will influence the

certainty of B, which then influences the certainty of C.

Similarly, evidence about C will influence the certainty of A through B.

If the state of B is known, then the channel is blocked, A and C become independent.

we say that A and C are d-separated given B.

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Diverging Connections Influence can pass between all

the children of A if A is not known. That is, B,C, . . . , E are d-separated given A.

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Sex (male, female),length of hair (long, short), and stature (<168 cm, ≥168 cm)

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Converging ConnectionIf nothing is known about A then

the parents are independent evidence about one of them cannot influence the certainties of the others through A.

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D-separationTwo distinct variables A and B in a

causal network are d-separated such that either:◦The connection is serial or diverging

and V is instantiated.◦The connection is converging, and

neither V nor any of V ’s descendants have received evidence.

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ExampleAre B and C independent given

A?

Are B and C independent given F

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Markov Blanket

The Markov blanket of a variable A is the set consisting of:◦the parents of A, ◦the children of A, and ◦the variables sharing a child with A.

The Markov blanket has the property that when instantiated, A is d-separated from the rest of the network.

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OutlineReasoning under uncertaintyCausal network and d-separation

Bayesian networkGraphical model

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Bayesian NetworkA Bayesian network consists of the

following◦A set of variables and a set of directed

edges between variables.◦Each variable has a finite set of mutually

exclusive states.◦The variables together with the directed

edges form an acyclic directed graph.◦To each variable A with parents B1, . . . ,

Bn, a conditional probability table P(A|B1, . . . , Bn) is attached.

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Bayesian Network The probabilities to specify are: P(A), P(B), P(C | A,B), P(E |C), P(D|C), P(F |E), and P(G| D,E,F)

It has been claimed that prior probabilities are bias to the model

Prior probabilities are necessary because prior certainty assessments are an integral part of human reasoning about certainty

The model should not include conditional independences that do not hold in the real world.

The d-separation properties check’s Conditional independences in model.

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Chain Rule for Bayesian NetworkLet BN be a Bayesian network

over U = {A1, . ..,An}. Then BN specifies a unique joint probability distribution P(U) given by the product of all conditional probability tables specified in BN:

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OutlineReasoning under uncertainty

Causal network and d-separation

Bayesian networkGraphical model

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Graphical ModelGraphical specification is easy for

humans to read, and helps focus attention.

The basic property of the Bayesian networks is the chain rule for compact representation of joint probability distribution.

Graphical model represents a causal relation in a knowledge domain.

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Questions

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